Factoring Special Forms

1
5.5
Factoring Special Forms
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Special Polynomials
In this section we will consider some polynomials
that have special forms that make it easy for us
to see how they factor. You may look at a
polynomial and say, Oh, thats just a difference
of squares or I think we have a sum of cubes
here. When you have a special polynomial, in
particular one that is a difference of two
squares, a perfect square polynomial, or a sum or
difference of cubes, you will have a factoring
formula memorized and will know how to proceed.
Thats why these polynomials are special.
They may just become our best friends among the
polynomials..
Blitzer, Algebra for College Students, 6e Slide
2 Section 5.5
3
The Difference of Two Squares
The Difference of Two Squares
If A and B are real numbers, variables, or algebraic expressions, then In words The difference of the squares of two terms, factors as the product of a sum and a difference of those terms.
Blitzer, Algebra for College Students, 6e Slide
3 Section 5.5
4
The Difference of Two Squares
EXAMPLE
Factor
SOLUTION
We must express each term as the square of some
monomial. Then we use the formula for factoring
Express as the difference of two squares
Factor using the Difference of Two Squares method
Blitzer, Algebra for College Students, 6e Slide
4 Section 5.5
5
The Difference of Two Squares
EXAMPLE
Factor
SOLUTION
The GCF of the two terms of the polynomial is 6.
We begin by factoring out 6.
Factor the GCF out of both terms
Factor using the Difference of Two Squares method
Blitzer, Algebra for College Students, 6e Slide
5 Section 5.5
6
The Difference of Two Squares
EXAMPLE
Factor completely
SOLUTION
Express as the difference of two squares
The factors are the sum and difference of the
expressions being squared
The factor is the difference of two
squares and can be factored
Blitzer, Algebra for College Students, 6e Slide
6 Section 5.5
7
The Difference of Two Squares
CONTINUED
The factors of are the sum and
difference of the expressions being squared
Thus,
Blitzer, Algebra for College Students, 6e Slide
7 Section 5.5
8
Factoring Completely
EXAMPLE
Factor completely
SOLUTION
Group terms with common factors
Factor out the common factor from each group
Factor out x 3 from both terms
Factor as the difference of two squares
Blitzer, Algebra for College Students, 6e Slide
8 Section 5.5
9
The Sum Difference of Two Cubes
Factoring the Sum Difference of Two Cubes
Factoring the Sum of Two Cubes Same Signs Opposite Signs
2) Factoring the Difference of Two Cubes Same Signs Opposite Signs
Blitzer, Algebra for College Students, 6e Slide
9 Section 5.5
10
The Sum Difference of Two Cubes
EXAMPLE
Factor
SOLUTION
Rewrite as the Sum of Two Cubes
Factor the Sum of Two Cubes
Simplify
Thus,
Blitzer, Algebra for College Students, 6e Slide
10 Section 5.5
11
The Sum Difference of Two Cubes
EXAMPLE
Factor
SOLUTION
Rewrite as the Difference of Two Cubes
Factor the Difference of Two Cubes
Simplify
Thus,
Blitzer, Algebra for College Students, 6e Slide
11 Section 5.5
12
5.5 Assignment
p.359 (2-18 even, 24-36 even, 76-84 even)